Fix a smooth manifold with a connection . Then parallel translation along a curve beginning at and ending at leads to an isomorphism , which depends smoothly on . For any , we get isomorphisms depending smoothly on . (Of course, given an isomorphism of vector spaces, there is an isomorphism sending —the important thing is the inverse.) (more…)

I’m going to try participating in Charles Siegel’s MaBloWriMo project of writing a short post a day for a month. In particular, I’m categorizing yesterday’s post that way too. I’m making no promises about meeting that every day, but much of the material I talk about lends itself to bite-sized pieces anyway.

There is a nice way to tie together (dare I say connect?) the material yesterday on parallelism with the axiomatic scheme for a Koszul connection. In particular, it shows that connections can be recovered from parallelism.

So, let’s pick a nonzero tangent vector , where is a smooth manifold endowed with a connection , and a vector field . Then makes sense from the axiomatic definition. We want to make this look more like a normal derivative.

[Nobody should read this post without reading the excellent comments below. It turns out that thinking more generally (via connections on the pullback bundle) clarifies things. Many thanks to the (anonymous) reader who posted them. -AM, 5/16]

A couple of days back I covered the definition of a (Koszul) connection. Now I will describe how this gives a way to differentiate vector fields along a curve.

Covariant Derivatives

First of all, here is a minor remark I should have made before. Given a connection and a vector field , the operation is linear in over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point can be defined if is replaced by a tangent vector at . In other words, we get a map , where denotes the space of vector fields. We’re going to need this below. (more…)